Hermitian skew Laplacian matrix of oriented graphs

Abstract

Let [Formula: see text] be a simple graph of order [Formula: see text] having [Formula: see text] edges and let [Formula: see text] be an orientation of [Formula: see text]. The hermitian skew Laplacian matrix [Formula: see text] of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix of oriented degrees [Formula: see text] and [Formula: see text] is the skew matrix. The spectral radius of the matrix [Formula: see text] is the hermitian skew Laplacian spectral radius and the sum of the absolute values of eigenvalues of [Formula: see text] is the hermitian skew Laplacian energy of [Formula: see text]. In this paper we study the hermitian spectral radius and hermitian skew Laplacian energy of [Formula: see text]. We obtain some new bounds for the hermitian skew Laplacian spectral radius and characterize the extremal oriented graphs attaining these bounds. Further, we obtain some bounds for the hermitian skew Laplacian energy and locate the extremal oriented graphs.